Off-diagonal Bethe Ansatz for the $D^{(1)}_3$ model

The exact solutions of the $D^{(1)}_3$ model (or the $so(6)$ quantum spin chain) with either periodic or general integrable open boundary conditions are obtained by using the off-diagonal Bethe Ansatz. From the fusion, the complete operator product identities are obtained, which are sufficient to enable us to determine spectrum of the system. Eigenvalues of the fused transfer matrices are constructed by the $T-Q$ relations for the periodic case and by the inhomogeneous $T-Q$ one for the non-diagonal boundary reflection case. The present method can be generalized to deal with the $D^{(1)}_{n}$ model directly.

The exact solution of integrable models associated with high rank Lie algebra is a very interesting issue. The case that the models with periodic or diagonal boundary reflection, where the U(1)-charge is conserved, has been studied extensively. For example, Reshetikhin derived the energy spectrum of the periodic quantum integrable models associated with B n , C n , D n and other Lie algebras by using the analytic Bethe Ansatz [17,18]. Martins and Ramos studied this kind of models with periodic boundary condition by using the algebraic Bethe Ansatz [19]. Li, Shi and Yue investigated the open boundary cases where the reflection matrices only have the diagonal elements [20,21].
The ODBA is a universal method to treat the quantum integrable models with or without U(1)-symmetry, even for the high rank cases. The nested ODBA was proposed to study the spin chain associated with A n Lie algebra with generic non-diagonal boundary reflections [22,23,24]. The ODBA also has been applied to the models associated with A (2) 2 [25], B 2 [26] and C 2 [27] Lie algebras.
In this paper, we develop a nested ODBA method to approach the quantum integrable D (1) 3 (simplest case of D (1) n ) model with either periodic or non-diagonal open boundary conditions. We have obtained the complete operator product identities based on the fusion [28,29,30,31] and the eigenvalues of the system based on the (inhomogeneous) T − Q relations.
The paper is organized as follows. In section 2, we review the R-matrix associated with the D

The R-matrix
Throughout, V denotes a six-dimensional linear space which endows the fundamental vectorial representation of so(6) (or the D 3 ) algebra, and let {|i , i = 1, 2, · · · , 6} be an orthogonal basis of it. We adopt the standard notations: for any matrix A ∈ End(V), A j is an embedding operator in the tensor space V ⊗ V ⊗ · · · , which acts as A on the j-th space and as identity on the other factor spaces; For B ∈ End(V ⊗ V), B ij is an embedding operator of B in the tensor space, which acts as identity on the factor spaces except for the i-th and j-th ones.
where the matrix elements are and u is the spectral parameter.
The R-matrix (2.1) satisfies the properties regularity : R vv where P 12 is the permutation operator with the matrix elements [P 12 ] ij kl = δ il δ jk , t i denotes the transposition in the i-th space, and R 21 = P 12 R 12 P 12 , and the crossing-matrix V is Combining the crossing-symmetry and the unitarity of the R-matrix, one can derive the crossing-unitarity relation crossing unitarity : The R-matrix (2.1) satisfies the Yang-Baxter equation

model with periodic boundary condition
For the periodic boundary condition, we introduce the monodromy matrix where the index 0 indicates the auxiliary space and the other tensor space V ⊗N is the physical or quantum space, N is the number of sites and {θ j } are the inhomogeneous parameters.
The monodromy matrix satisfies the Yang-Baxter relation The transfer matrix is given by the trace of monodromy matrix in the auxiliary space From the Yang-Baxter relation (3.2), one can prove that the transfer matrices with different spectral parameters commute with each other, [t (p) (u), t (p) (v)] = 0. Therefore, t (p) (u) serves as the generating function of the conserved quantities of the system. The Hamiltonian is given in terms of the transfer matrix (3.3) as where 5) and the periodic boundary condition reads

The fusion
At some special points, the R-matrix (2.1) degenerates into the projector operators which enables us to do the fusion. For example, at the point of u = −2, we have 8) and the basis vector reads From the Yang-Baxter equation (2.5), we obtain the following fusion identities In Eq.(3.11), we take the fusion in the auxiliary spaces and obtain a 16-dimensional auxiliary space V 12 . We can show that the resulting 16-dimensional auxiliary space is exact a direct tensor-product of two 4-dimensional auxiliary spaces V (s + ) and V (s − ) , which endows the (anti) spinor representation of so(6) respectively. The matrix S 1 ′ 2 ′ is defined in the direct The relation between S 1 ′ 2 ′ andS 1 ′ 2 ′ is provided by (4.19). In Eq.(3.11), the matrices R s ± v 1 ′ 2 (u) are defined in the V (s ± ) ⊗ V space and take the forms of where the matrix elements are The matrices R crossing unitarity : R (3.12) and satisfy the Yang-Baxter equations Now, we consider the degenerate case of matrix R where S (+) 1 ′ 2 is a constant matrix omitted here and P (+) with the basis vectors Taking the fusion by using the projector P (+) We note that the dimension of the fused auxiliary space with the basis vectors and the S (−) 1 ′ 2 is a constant matrix omitted here. Taking the fusion by using the projector P

The operator product identities
From the fused R-matrices R s ± v 0 ′ j , we can define the fused monodromy matrices which satisfy the Yang-Baxter relations Taking the partial trace in the auxiliary space, we obtain the fused transfer matrices t (p) From the Yang-Baxter relations (3.2) and (3.19) at certain points and using the properties of projectors, we obtain By using the fusion identities (3.9), (3.11), (3.15) and (3.17), we have following fusion iden- Taking the partial trace in the auxiliary spaces and using the relations (3.21) and (3.22), we obtain following operator product identities From the definitions, the asymptotic behaviors of the transfer matrices can be calculated directly as Let us denote the eigenvalues of the transfer matrices t (p) (u) and t (p) ± (u) as Λ (p) (u) and Λ (p) ± (u), respectively. We note that the eigenvalues Λ (p) (u) and Λ And the corresponding asymptotic behaviors read Then we arrive at that 4N functional relations (3.25) together with 3 asymptotic behaviors (3.26) give us sufficient conditions to determine the eigenvalues.

The T − Q relations
Let us introduce some functions The relations (3.25) and (3.26) enable us to parameterize the eigenvalues of the transfer matrices in terms of some homogeneous T − Q relations as . (3.28) The regularities of the eigenvalues Λ (p) (u) and Λ We note that the BAEs obtained from the regularity of Λ (p) (u) are the same as those obtained from the regularity of Λ (p) ± (u). It is easy to check that Λ (p) (u) and Λ (p) ± (u) satisfy the functional relations (3.25) and the asymptotic behaviors (3.26). Therefore, we conclude that Λ (p) (u) and Λ (p) ± (u) are the eigenvalues of the transfer matrices t (p) (u) and t (p) ± (u), respectively. It is easy to check that the homogeneous T − Q relations (3.28) and associated BAEs (3.29) coincide with those obtained previously by others methods [17,18,19].
Then the eigenvalues of the Hamiltonian (3.4) can be obtained by the Λ (p) (u) as

model with non-diagonal open boundary condition
In this section, we consider the system with general integrable open boundary condition.
Let us introduce a pair of reflection matrices K v (u) andK v (u). The former satisfies the reflection equation (RE) and the latter satisfies the dual RE For the open case, instead of the "row-to-row" monodromy matrix T v 0 (u) given by (3.1), one needs consider the "double-row" monodromy matrix to describe the reflection process. Let us introduce another "row-to-row" monodromy matrix which satisfies the Yang-Baxter relation The transfer matrix t(u) is defined as where H k k+1 is given by (3.5) and K v (0) = ξ × id.

Reflection matrix
In this paper, we consider the general integrable open boundary condition where the reflection matrix has the non-diagonal elements which breaks the U(1)-symmetry of the system. The general reflection matrix for D (1) n vertex model with its R-matrix being trigonometric has been obtained by Lima-Santos and Malara [34,35]. Here we chose the reflection matrix K v− (u), which is the solution of reflection equation (4.1), as where the non-vanishing matrix elements are Here c 1 , c 2 and c 3 are free boundary parameters, where c is expressed in terms of them as The Eq.(4.7) can also be obtained at some special free parameters by taking rational limit to the general reflection matrix in [34,35].
The dual reflection matrixK v (u) is also a non-diagonal one and given bȳ where c ′ 1 , c ′ 2 and c ′ 3 are the boundary parameters at the other side. For a generic choice of the three boundary parameters {c i , c ′ i |i = 1, 2, 3}, it is easily to check that [K v (u),K v (v)] = 0. This implies that the matrices K v (u) andK v (u) cannot be diagonalized simultaneously.
Following the method developed in [32] and using the crossing-symmetry (2.2) of the R-matrix, the explicit expressions (4.7) and (4.9) of the reflection matrices, we find that the transfer matrix (4.5) possesses the crossing symmetry t(−u − 2) = t(u). (4.10)

Fusion of the reflection matrix
In order to seek the relations that the transfer matrix (4.5) satisfies, we first consider the fusion of the reflection matrix. The fusion of the one-dimensional projector P vv(1) 12 gives P vv(1) 12 The fusion of the 16-dimensional projector P vv(16) 12 gives P vv(16) 21 and crossing unitarity : R (4.16) and satisfies the Yang-Baxter equations (4.17) The matrices K s ± (u) satisfy the reflection equations TheS 1 ′ 2 ′ can be obtained byS The dual reflection matricesK s ± (u) are constructed as 20) which satisfy the dual REs The fusion of the 4-dimensional projector P (±)

Operators product relations
Define the fused monodromy matricesT ± 0 ′ (u) by using the fused R-matrix R vs ± aŝ which satisfy the Yang-Baxter relation From the Yang-Baxter relations (4.4) and (4.24) at certain points and using the properties of projectors, we obtain By using the fusion identities (3.9), (3.11), (3.15) and (3.17), we also obtain the fusion identities P vv(1) The fused transfer matrices are defined as Direct calculation shows Meanwhile, from the definitions we also know the values of transfer matrices at some special points The corresponding asymptotic behaviors read Denote the eigenvalues of the fused transfer matrices t(u) and t ± (u) as Λ(u) and Λ ± (u), respectively. From Eq.(4.31 ), we obtain the functional relations among the eigenvalues of the transfer matrices Eqs. (4.32) and (4.33) give rise to the relations and

Inhomogeneous T-Q relations
Let us introduce some functions k + 1), k + 1), The constraints (4.34)-(4.36) enables us to parameterize the eigenvalues of the transfer matrices Λ(u) and Λ ± (u) in terms of the inhomogeneous T − Q relations .
The regularities of the eigenvalues Λ(u) and Λ ± (u) lead to that the Bethe roots {µ k + 1 2 ), k = 1, · · · , L 1 , l + 1 2 , l = 1, · · · , L 2 , where the numbers of Bethe roots should satisfy the constraint and the parameter x is given by Again, the BAEs obtained from the regularity of Λ(u) are the same as those obtained from the regularity of Λ ± (u). The function Q (m) (u) has two zero points, and the BAEs obtained from these two points are the same. It is easy to check that Λ(u) and Λ ± (u) satisfy the functional relations (4.34), the values at the special points (4.35) and the asymptotic behaviors (4.36).
Therefore, we conclude that Λ(u) and Λ ± (u) are the eigenvalues of the transfer matrices t(u) and t ± (u), respectively.
Finally, the eigenvalue E of Hamiltonian (4.6) can be obtained by the Λ(u) as (4.41)

Discussion
In this paper, we study the exact solution of D 3 model, with various boundary conditions including the periodic one and the non-diagonal reflection one. By using the fusion technique, we obtain the complete operator product identities of the fused transfer matrices. Based on them and the asymptotic behaviors as well as the special values at certain points, we obtain the Bethe Ansatz solutions of the system. The method and the results in this paper could be generalized to the D

Appendix A: Spinorial R-matrix
In this Appendix, we show that the vectorial R-matrix R vv (u) (2.1) and the fused ones R s ± v (u) (see above (3.12)) can be obtained from the spinorial R-matrix of D (1) 3 model [33] by using the fusion. The spinorial R-matrix R ss 1 ′ 2 ′ (u) of the D 3 model is the fundamental R-matrix of su(4) one, and is a 16 × 16 matrix with the form The spinorial R-matrix has following properties unitarity : R ss 1 ′ 2 ′ (u)R ss 2 ′ 1 ′ (−u) = (1 + u)(1 − u), crossing − unitarity : R ss and satisfies the Yang-Baxter equation At the point of u = −1, we have R ss 1 ′ 2 ′ (−1) = P where P 1 ′ 2 ′ is a 6-dimensional projector The dimension of the fused space V 1 ′ 2 ′ is 6. From Eq.(A.5), we obtain the fused R-matrix R s + v (u). The R s − v (u) can be obtained via Eq. For the open case, the spinorial R-matrix R ss (u) and the spinorial reflection matrix K s (u) satisfy the reflection equation One can check that matrix (4.14) is a solution of Eq.(A.7), K s + (u) = K s (u). By using Eq.(4.22), we arrive at K s − (u) (4.15). The vectorial reflection matrix K v (u) (4.8) is obtained from K s + (u) by using the fusion with 6-dimensional projector P